In many MR-applications it is useful to be able to shorten the excitation time and begin acquiring a signal as soon after the RF-excitation as possible. In particular, one would like to reduce the rephasing time, essentially to 0. This is especially important when imaging very short T2-species. The possibility of using two “half pulses” is explored by Nielson et al. in “Ultra-short echo-time 2D time-of-flight MR angiography using a half-pulse excitation,” Mag. Res. in Medicine, Vol. 41 (1999), pp. 591–599. The idea is to use two excitations, neither of which requires rephrasing, so that the summation of these two independent measurements equals the result of using a single excitation, which would require rephasing.
More precisely, Nielson et al. consider whether, given a summed selective excitation transverse profile, herein referred to as a target transverse profile (mx+imy)(ƒ), one can find a pair of self refocused pulses q1 and q2 with transverse magnetization profiles, (mx1+imy1)(ƒ), (mx2+imy2)(ƒ), so that:(mx+imy)(ƒ)=[(mx1+imy1)(ƒ)+(mx2+imy2) (−ƒ)].  (1)By “self refocused” it is meant that, at the conclusion of the RF-pulse, the magnetization achieves the stated transverse profile, without any need for rephasing. The pair (q1, q2) are called a pair of half pulses.
Assuming the possibility of instantaneously switching the gradient fields, such a pair of half pulses would be used in an MR-experiment as follows. With the slice select gradient turned on, the sample is first excited using the profile q1. At the conclusion of the excitation, the slice select gradient is immediately turned off, and the signal S1(t) is acquired. Once the system has returned to equilibrium, the slice select gradient is again turned on, but with the polarity reversed. The sample is excited using the profile q2. At the conclusion of this excitation, the slice select gradient is again immediately turned off, and the signal S2(t) is acquired.
Let q denote the (minimum energy) pulse which, after appropriate rephasing produces the target excitation. Ignoring noise and relaxation effects, it is not difficult to see that if the transverse magnetizations satisfy Equation (1), then the sum of the signals (S1+S2) equals the signal that would be measured if one used a single excitation using the profile q, and started acquiring the signal after the appropriate rephasing. In light of the linearity of the measurement process, it is also clear that one can follow the excitations with phase, or frequency encoding steps, and the statement about the sum of the measured signals remains correct.
Due to the nonlinear dependence of the magnetization profile on the pulse envelope, the problem of finding the pair (q1, q2), given mx+imy, is obviously nonlinear. The Nielson paper solves this problem to first order, using the low flip angle connection between the Fourier transform of the pulse envelope, and the transverse magnetization profile. This solution is adequate for very small target flip angles; however, for larger flip angles, the linear approximation has rather poor selectivity. An example of using the linear approximation to design a pair of half pulses which sum to produce a 90° flip as the summed target magnetization is shown in FIG. 1, where FIG. 1(a) illustrates half of a symmetric 90° pulse and FIG. 1(b) illustrates the transverse magnetization produced by using the pulse in FIG. 1(a) as a half pulse.
As will be explained in more detail below, the present inventors have found a way to reformulate the half pulse synthesis problem and exactly solve it using the inverse scattering mathematical formalism. In the natural time parameterization provided by the inverse scattering formalism, a pulse is self refocused if it is supported in (−∞,0], as noted by Rourke et al. in “The inverse scattering transform and its use in the exact inversion of the Bloch equation for noninteracting spins,” Jour. of Mag. Res., Vol. 99 (1992), pp. 118–138, and by Epstein in “Minimum Power Pulse Synthesis Via the Inverse Scattering Transform,” Jour. of Mag. Res., Vol. 167 (2004), pp. 185–210. In this time parameterization, the Fourier transforms of q1 and q2 would therefore have analytic continuations to the upper half plane. From the mathematical standpoint, the content of the Nielson paper is contained in the following classical theorem:
Theorem 1. Let ƒεL2(R), then there are unique functions ƒ1, ƒ2 in L2(R) such that ƒ1and ƒ2 have analytic extensions to the upper halfplane and ƒ (ξ)=ƒ1(ξ)+ƒ2(−ξ).
In the Nielson paper, various practical difficulties with implementing half pulses are discussed. For example, the pulses produced by either the linear or nonlinear theory produce considerable excitation outside the desired slice. The selectivity of the pair of pulses results from delicate cancellations between the out-of-slice contributions from the two excitations. A variety of phenomena, such as eddy currents, can lead to imperfect cancellation out-of-slice, in the sum of the measured signals. To attain a high degree of cancellation, Nielson, et al. found it necessary to measure the actual gradient fields, with the sample in place. Nielson, et al. then use a VeRSE technique to match the play out of the half pulses to the actual time course of the gradient.
The present invention provides a solution to the half pulse synthesis problem so that an arbitrary, admissible target transverse profile is producible as the summed response to two, self refocused selective “half pulse” excitations may be obtained for use in, e.g., magnetic resonance imaging pulse generation.